Parametrizing the infinite dimensional parameter \(f_0\)  in (REF ) using the multinomial model with unknown probabilities at the observations results in the maximum empirical likelihood estimator (MELE) [1]} \(\tilde{\mathbf {\alpha }}\)  of \(\mathbf {\alpha }\)  and step-function estimator of \(f_0\) .
The MELE \(\tilde{\mathbf {\alpha }}\)  can also be obtained by fitting the data \((X,D)\)  with the logistic regression (REF ). This method works well when \(f_0\)  is a nuisance parameter. However in many applications, both \(\mathbf {\alpha }\)  and \(f_0\)  are of interest. A jagged step-function estimate of \(f_0\)  is unsatisfactory especially when sample sizes are small. Smooth and efficient estimator is desirable.
[2]} proposed to smooth the discrete empirical density estimates of \(f_0\)  and \(f_1\)  using kernel method. As a smoothing technique kernel density does not target the unknown density but its convolution with the scaled kernel function for any positive bandwidth. Good density estimation is key to solve many difficult statistical problems such as the goodness-of-fit test [3]} and the estimation of the receiver
operating characteristic curve when result of diagnostic test is continuous [4]} and sample size is small.
In this paper, we shall investigate the estimation of densities and the parameters under model (REF ) using approximate Bernstein likelihood method.
